Portfolio Optimization


MGMT 675
AI-Assisted Financial Analysis
Kerry Back

Objectives

  • Frontier portfolios of risky assets (minimize risk subject to achieving a target expected return)
  • Tangency portfolio (maximize Sharpe ratio)
  • Estimate means, standard deviations, and correlations from historical returns
  • Download adjusted closing prices from Yahoo Finance and compute returns

Goal Seek in Python

  • There are several choices in python for optimizing functions.
  • The qp function from cvxopt is a very good choice for portfolio optimization.
    • cvxopt = convex optimization
    • qp = quadratic programming

Examples

  • From Applied:
    • U.S., U.K., France, Germany, Japan
    • U.S., Developed, Emerging
  • ETFs from Yahoo Finance

U.S., U.K., France, Germany, and Japan

  • Download international.xlsx and international_corrs.xlsx from the course website.
  • Upload the files to Julius. Ask Julius to read them.
  • Ask Julius to convert the means, standard deviations, and correlation matrix into numpy arrays.
  • Ask Julius to compute the covariance matrix as a numpy array.

  • Ask Julius to use cvxopt to minimize variance subject to achieving a target expected return.
    • If allowing short sales, tell Julius there are no inequality constraints.
    • If excluding short sales, tell Julius to not allow short sales.
  • Ask Julius to repeat for a range of target expected returns and to plot the expected returns and standard deviations.

Digression on cvxopt

Frontier Portfolios Allowing Short Sales

  • minimize \((1/2) x'Px\) subject to \(Ax=b\)
  • \(P=\) covariance matrix. \(x'Px\) is portfolio variance
  • \(A =\) array with two rows.first row \(=\) np.ones(n).second row \(=\) asset expected returns.
  • \(b=\) np.array([1, targ])
  • \(Ax=b\) means weights sum to 1 and expected return \(=\) targ.
  • Julius should figure all of this out.

Frontier Portfolios Excluding Short Sales

  • minimize \((1/2) x'Px\) subject to \(Ax=b\) and \(Gx \le h\)
  • \(P\), \(A\), and \(b\) as before
  • \(G =-\) np.eye(n) and \(h =\) np.zeros(n)
  • \(Gx \le h\) means weights are nonnegative (no shorts)
  • Again, Julius should figure this out.

Back to Example

Tangency Portfolio

  • Give Julius a number for the risk-free rate.
  • Ask Julius to minimize the variance minus the risk premium.
    • If allowing short sales, tell Julius there are no inequality constraints.
    • If excluding short sales, tell Julius to not allow short sales.
  • Ask Julius to divide by the sum of the weights to compute the tangency portfolio.
  • Ask Julius to include the tangency portfolio and the capital market line on the previous plot.

Another Digression on cvxopt

cvxopt for Tangency Portfolio

  • Minimize \(x'Px - q'x\)
    • \(P=\) 2 times covariance matrix (but the 2 is not important)
    • \(q=\) risk premia
  • No equality constraints
  • If no short sales, then \(G=-\) np.eye(n) and \(h=\) np.zeros(n)
  • Then divide by the sum of weights.

US, Developed, and Emerging

  • Start a new chat.
  • Upload us_developed_emerging_rets.xlsx and ask Julius to read it.
  • Ask Julius to compute the sample means, sample standard deviations and sample correlation matrix as numpy arrays.
  • Repeat the frontier and tangency portfolio calculations.

Data from Yahoo

ETFs from Yahoo

  • Example: ask Julius to use yfinance to get Yahoo adjusted closing prices for
    • SPY = S&P 500
    • VBR = Vanguard small-cap value
    • IEF = Treasury bonds
    • UUP = U.S. dollar bullish
  • Ask Julius to downsample prices to end-of-month and compute monthly returns as percent changes in the downsampled prices.

  • Ask Julius to compute means, standard deviations, and correlation matrix as numpy arrays.
  • Ask Julius to find frontier of risky assets and tangency portfolio as before.